linewidths in spectroscopy

for an absorption from E₀ to E_1, what are all the terms?

N₀ and N₁ are populations in state 0 and state 1

B_1←0 is the rate constant for absorption from state 0 to 1

p_𝜈(𝜈₁₀) is the radiation density at the precise frequency 𝜈₁₀ needed to drive a transition from ground to excited state

for an emission from E₁ to E_0, what are all the terms?

N₁ is the populations in state 1

A_1→0 is the rate constant for the spontaneous emission from state 1 to 0

show stimulated emission diagram from E₁ to E₀

hint : laser amplification

resulting photons are exactly in phase so they add

for stimulated emission from E₁ to E₀, what are all the terms?

N₁ is the populations in state 1

B_1→0 is the rate constant for the stimulated emission from state 1 to 0

p_𝜈(𝜈₁₀) is the radiation density at the precise frequency needed to drive a transition from excited to ground state

how are these three equations related to the lifetime?

• a measurement of the absorption process can be used to determine the lifetime of the excited state

how are the Einstein coefficients related at equilibrium?

how is the width of a frequency peak related to the lifetime of excited state?

more time spent in excited state means a narrower peak

show time decay (for a greater vs smaller amount of time) vs resulting peak shapes

what is the name of this type of peal?

Lorentzian peak shape

what is the time energy uncertainty principle?

what is the linewidth the same as?

what happens when the lifetime changes?

linewidth is that of a damped oscillator

homogenous broadening

show doppler broadening diagram

what is the equation for frequency of approaching?

s is velocity of molecule

c is speed of light

𝜈₀ is vibrational frequency

what is the equation for frequency of receding?

how is a spectra related to Lorentzian peaks?

it is a sum of homogenously broadened Lorentzian peaks

when is the spectra Lorentzian?

when there is a series of molecule travelling slightly faster/slower

• gives normal distribution

when does the spectra become Gaussian?

when the transition frequency varies because of many small independent perturbations

how is the Doppler effect an example of this? when is this dominant?

inhomogeneous broadening

• dominant at low P, high T and high 𝜈

any time the transition frequency varies because of many small independent perturbations, peak shape becomes Gaussian

in gases, what is the relation between collision lifetime and frequency?

as collision frequency increases, time in excited state decreases

what are the terms?

when is pressure broadening in a gas significant? what is the spectra peak shape?

significant at high P, low T, low 𝜈

example of homogenous broadening, linewidth is that of damped oscillator = Lorentzian peak shape